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The expression provided is:

\[
\frac{x^2 + 5x - 7}{(x - 1)(x^2 - 3)}
\]

To resolve this into partial fractions, we express it in the form:

\[
\frac{x^2 + 5x - 7}{(x - 1)(x^2 - 3)} = \frac{A}{x - 1} + \frac{Bx + C}{x^2 - 3}
\]

### Steps to Solve:

1. **Multiply both sides by the denominator** \((x - 1)(x^2 - 3)\) to eliminate the fractions:

   \[
   x^2 + 5x - 7 = A(x^2 - 3) + (Bx + C)(x - 1)
   \]

2. **Expand the right-hand side**:

   \[
   x^2 + 5x - 7 = A(x^2 - 3) + (Bx + C)(x - 1)
   \]
   \[
   = Ax^2 - 3A + Bx^2 - Bx + Cx - C
   \]
   \[
   = (A + B)x^2 + (-B + C)x + (-3A - C)
   \]

3. **Equate the coefficients of corresponding powers of \(x\) from both sides**:

   - Coefficient of \(x^2\): \( A + B = 1 \)
   - Coefficient of \(x\): \( -B + C = 5 \)
   - Constant term: \( -3A - C = -7 \)

4. **Solve the system of equations**:

   - From \( A + B = 1 \), we have \( B = 1 - A \).
   - Substitute \( B = 1 - A \) into \( -B + C = 5 \):
     \[
     -(1 - A) + C = 5 \Rightarrow -1 + A + C = 5 \Rightarrow A + C = 6 \Rightarrow C = 6 - A
     \]
   - Substitute \( C = 6 - A \) into \( -3A - C = -7 \):
     \[
     -3A - (6 - A) = -7 \Rightarrow -3A - 6 + A = -7 \Rightarrow -2A - 6 = -7 \Rightarrow -2A = -1 \Rightarrow A = \frac{1}{2}
     \]
   - With \( A = \frac{1}{2} \), \( B = 1 - A = \frac{1}{2} \), and \( C = 6 - A = \frac{11}{2} \).

5. **Substitute back into the partial fraction form**:

   \[
   \frac{x^2 + 5x - 7}{(x - 1)(x^2 - 3)} = \frac{\frac{1}{2}}{x - 1} + \frac{\frac{1}{2}x + \frac{11}{2}}{x^2 - 3}
   \]

   To simplify the fractions further:

   \[
   = \frac{1}{2}\left(\frac{1}{x - 1} + \frac{x + 11}{x^2 - 3}\right)
   \]

This is the partial fraction decomposition of the given expression.

Would you like any further details or have any questions?

Here are some related questions you might find interesting:

1. What if the quadratic factor in the denominator had complex roots?
2. How would the decomposition change if there were repeated linear factors?
3. Can partial fractions be used to simplify the integration of the expression?
4. What if the numerator had a higher degree than the denominator?
5. How do you handle partial fractions when there are irreducible quadratic factors in the denominator?

**Tip:** Always simplify the fractions where possible after performing the partial fraction decomposition for a cleaner and more interpretable result.

Learn how to resolve partial fraction decomposition for the expression (x^2 + 5x - 7) / ((x - 1)(x^2 - 3)). This problem involves advanced algebraic techniques, including partial fraction decomposition and polynomial division. Suitable for advanced high school students.

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